Role of back-diffusion studied by computer simulation
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I. INTRODUCTION
SOLIDIFICATION of alloys generally occurs with segregation, and this phenomenon may have a strong effect on the progress of the solidification. There has, thus, been a considerable interest in modeling segregation. One must distinguish between two types. The present work only concerns microsegregation, whereas the effects of macrosegregation will be completely neglected. The simplest case of microsegregation is modeled by considering a small part of a dendrite and the surrounding liquid. Often one applies a planar model, which is an approximation but may be justified as a reasonable simplification of a planar arrangement of a set of densely spaced secondary arms. Another approximation may be to model the thickening of a single secondary arm by considering a cylindrical geometry. More ambitious treatments of the geometry of dendrites, including coarsening, have been attempted by Sundarraj and Voller,[1] for instance, but will not be discussed further in the present article. A crude but very powerful method of estimating microsegregation was proposed by Gulliver in 1913[2] and was again applied by Scheil in 1942.[3] The main approximations were to treat the rate of diffusion in the liquid as infinite but negligible in the solid phases. With that model, one does not have to specify the geometry. An additional approximation was to treat the liquidus and solidus in the phase diagram as straight lines. That approximation is no longer so important, due to the high speed of numerical computer calculations. Computer programs for the simulation of microsegregation in alloys are now common. They can be coupled to thermodynamic databases from which the solid/ liquid equilibrium can be calculated repeatedly during the simulation. They can, thus, be used for alloys with a nonlinear solidus and liquidus and with more than two components (for example, Reference 4). It was recently shown[5] that a simulation program can even be obtained directly from a thermodynamic databank with facilities for stepwise calculations of equilibria. Only minor modifications were required. Another important improvement was the introduction of ยจ M. HILLERT, Professor, L. HOGLUND, Research Associate, and M. SCHALIN, Graduate Student, are with the Department of Materials Science and Engineering, KTH,SE-10044 Stockholm, Sweden. Manuscript submitted September 15, 1998. METALLURGICAL AND MATERIALS TRANSACTIONS A
more realistic treatments of diffusion, which was done by modifying the analytical expression of Gulliver and Scheil. It was first done by Broady and Flemings,[6] and their treatment was later improved by Cline and Kurz[7] and Ohnaka.[8] The resulting equations are simple to use, but Kobayashi[9] finally gave an exact treatment of the model proposed by Broady and Flemings and found that all three equations proposed by the previous authors could sometimes give very erroneous results. Even though Kobayashi gave an exact treatment, it must be realized that the basic assumptions are not very good. For instance, a parabolic growth law wa
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